Meg decides to throw her 10-year-old daughter Sue a birthday party and allows her to invite 15 friends to the house. Meg purchases 25 red balloons for the children, since red is Sue\'s favorite color. In how many ways can the balloons be distributed to the children at the birthday party (including Sue), assuming that no child should go home without a balloon? An English professor has chosen five novels for a course in contemporary literature. Each student must choose to write an analysis of at least one of the novels. Extra credit will be given for every additional analysis. There are six students enrolled in the class. What is the number of ways the students can submit reports? I have a bowl of identical candy bars. Four young trick-or-treaters have just knocked on my door and are holding their bags out for some candy. If I grab six candy bars from my bowl, in how many ways can I distribute all six of them? Give two answers: (a) I will try to be fair (b) I need not be fair. The people in charge of a raffle are going to give out four prize bags to the raffle winners. They want to make sure that each prize bag contains at least one of the 11 gift certificates for a department store. In how many ways can the prize bags be prepared, assuming that all the gift certificates are of the same monetary value?
1) No of balloons = 25
No of children including Sue = 15+1 =16
Since atleast one ball must be given to one child first 16 balls (identical) give to 16 children each.
This can be done in 1 way.
Now remaining balls are 9.
I ball can be given to any one of the 16,
II also to any of the 16
...
9th also to any of the 16
Hence total number of ways to distribute the balloon = 1X16x16x... 9 times
= 169 times
= 68719476736
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2) No of novels =10
No of student =6
Since each student has to submit atleast one report on novel, any 6 novels will be selected by 6 students.
REmaining 4 novels can be selected by any of the 10 in any way
So no of ways = 1x10x10x10x10 = 10000 ways
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3) No of children = 4
No of candies = 6
a) To be fair:
Each child must get atleast one.
Hence first 4 each to one child. Remaining 2 to any of the 4 children
No of ways = 1x4x4 = 16 ways
b) Not fair
A child may get all 6 or a child may not get anything.
Hence I candy to any of the 6, ii also to any of the 6,....
No of ways=64=1296
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